Exam 16: Integrating Functions of Several Variables

Evaluate the integral $\int _ { - 2 } ^ { 0 } \int _ { - 2 } ^ { 0 } \int _ { - \sqrt { 4 - x ^ { 2 } } } ^ { \sqrt { 4 - x ^ { 2 } } } \left( x ^ { 2 } + z ^ { 2 } \right) ^ { 7 / 2 } d z d x d y$ in cylindrical coordinates.

The region W is shown below.Write the limits of integration for $\int _ { w } f ( x , y , z ) d V$ in spherical coordinates.

Evaluate exactly the integral $\int _ { R } y d A$ , where R is the region shown below.

Set up an iterated integral for $\int _ { W } f ( x , y , z ) d V$ , where W is the solid region bounded below by the rectangle 0 $\le$ x $\le$ 3, 0 $\le$ y $\le$ 1 and above by the surface $z ^ { 2 } + y ^ { 2 } = 1$

Evaluate $\int _ { - 1 } ^ { - 1 } \int _ { \sqrt { 1 - y ^ { 2 } } } ^ { 0 } \frac { 9 \sqrt { x ^ { 2 } + y ^ { 2 } } } { 1 + x ^ { 2 } + y ^ { 2 } } d x d y$ Provide an exact answer.

Let W be the region between the spheres $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ .Given that $\int _ { W } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 1 / 2 } d V = 15 \pi$ , evaluate the integral $\int _ { w } \left( 64 x ^ { 2 } + 36 y ^ { 2 } + 144 z ^ { 2 } \right) ^ { 1 / 2 } d V$ , where $\bar{W}$ is the region between the ellipsoids $\frac { x ^ { 2 } } { 3 ^ { 2 } } + \frac { y ^ { 2 } } { 4 ^ { 2 } } + \frac { z ^ { 2 } } { 2 ^ { 2 } } = 1$ and $\frac { x ^ { 2 } } { 3 ^ { 2 } } + \frac { y ^ { 2 } } { 4 ^ { 2 } } + \frac { z ^ { 2 } } { 2 ^ { 2 } } = 4$ .

Estimate $\int$ _R f(x, y)dA using the table of values below, where R is the rectangle 0 $\le$ x $\le$ 4, 0 $\le$ y $\le$ 6 y x 0 3 6 0 2 3 4 2 6 4 3 4 18 14 12

Consider the change of variables x = s + 3t, y = s - 2t. Let R be the region bounded by the lines 2x + 3y = 1, 2x + 3y = 4, x - y = -3, and x - y = 2.Find the region T in the st-plane that corresponds to region R. Use the change of variables to evaluate $\int _ { R } 2 x + 3 y d A$

Find the mass of the solid cylinder $x ^ { 2 } + y ^ { 2 } \leq 25 , \quad 4 \leq z \leq 5$ with density function $f ( x , y , z ) = z + 4 \left( x ^ { 2 } + y ^ { 2 } \right)$

Let x and y have joint density function $p ( x , y ) = \left\{ \begin{array} { l l } x + y & \text { if } 0 \leq x \leq 1,0 \leq y \leq 1 \\0 & \text { otherwise }\end{array} \right.$ Find the probability that 0.5 $\le$ x $\le$ 0.6.

Reverse the order of integration for the following integral. $\int _ { 1 } ^ { 3 } \int _ { x ^ { 2 } - 9 } ^ { 4 x - 12 } f ( x , y ) d y d x$

Let R be the ice-cream cone lying inside the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and inside the cone $z = \sqrt { 3 \left( x ^ { 2 } + y ^ { 2 } \right) }$ .Find the center of mass of R.

Jane and Mary will meet outside the library at noon.Jane's arrival time is x and Mary's arrival time is y, where x and y are measured in minutes after noon.The probability density function for the variation in x and y is $p ( x , y ) = \frac { 1 } { 180 } e ^ { - x / 12 } e ^ { - y / 15 }$ After Jane arrives, she will wait up to 15 minutes for Mary, but Mary won't wait for Jane.Find the probability that they meet.

An arrow strikes a circular target at random at a point (x, y), using a coordinate system with origin at the center of the target.The probability density function for the point where the arrow strikes is given by $p ( x , y ) = \frac { 1 } { \pi } e ^ { - x ^ { 2 } - y ^ { 2 } }$ What is the probability that the arrow strikes within 0.45 feet of the center of the target.

Calculate the following integral: $\int _ { R } r ^ { 5 } \cos \theta d A$ where R is the shaded region shown below. ·

Find the condition on the non-negative constants a and b for p(x, y)to be a joint density function, where $p ( x , y ) = \left\{ \begin{array} { c c } a x + b y & \text { if } 0 \leq x \leq 6,0 \leq y \leq 6 \\0 & \text { otherwise }\end{array} \right.$

Choose the most appropriate coordinate system and set up a triple integral, including limits of integration, for a density function f(x, y, z)over the given region.

Evaluate $\int _ { 0 } ^ { \pi } \int _ { x } ^ { \pi } \left( \frac { \sin 2 y } { y } - \sin 2 y \right) d y d x$ by first reversing the order of integration.

Find the volume of the solid bounded by the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ and the plane z = 1.

Evaluate the iterated integral. $\int_{1}^{e} \int_{0}^{1 / x} \cos (4 x y) d y d x$